class: center, middle, inverse, title-slide # Lecture 5 ## The Effects Model ### Psych 10 C ### University of California, Irvine ### 04/08/2022 --- ## Null Model - Last class we used mathematical notation and the normal distribution to represent what we called the **Null Model** -- - This model assumed that all `\(i = 1,\dots, 4\)` observed participants of the `\(j=1,2\)` groups where samples from the same distribution. -- - In other words, it formalizes our verbal hypothesis that there are no differences between groups. -- - The **Null Model** is defined as: `$$y_{ij} \sim \text{Normal}(\mu, \sigma^2)$$` -- - Finally, we said that given that we don't know the values of our parameters `\(\mu\)` and `\(\sigma^2\)` which completely define the Normal distribution, we needed to infer (learn) them from our observations. -- - This is called **Statistical Inference**, but how can we actually get those numbers? --- ## Statistical Inference for the Normal distribution - One of the key advantages of the Normal distribution is that its parameters are directly associated with the expectation and the variance of a random variable. -- - Our model assumes that our observations are random variables that follow a Normal distribution with parameters `\(\mu\)` and `\(\sigma^2\)`. -- - If a random variable `\(y\)` follows a Normal distribution with parameters `\(\mu\)` and `\(\sigma^2\)`, then we know that the following two statements are **TRUE**: `$$\mathbb{E}(y) = \mu$$` and `$$\mathbb{V}ar(y) = \sigma^2$$` -- - Remember that we already have a very good approximation to both `\(\mathbb{E}(y)\)` and to the `\(\mathbb{V}ar(y)\)`! --- class: inverse, middle, center # Estimators --- ## Estimators - In week 1 we talked about statistics as functions of our observations. -- - A statistic that is used to approximate the parameter (like `\(\mu\)`) in a statistical model is called an **estimator**. -- - Given that we know that our best statistic for the expected value of a r.v. is the mean or average of our observations, we can use it as an **estimator** for `\(\mu\)`. -- - We denote those estimators by adding a "hat" on top of the Greek character: `$$\hat{\mu}_0 = \frac{1}{n} \sum_{i} \sum_{j} y_{ij}$$` -- - Here, `\(n\)` represents the total number of observations that we added to calculate the mean, and the indices `\(i\)` and `\(j\)` denote the observation number and the group respectively. --- ## Variance estimator - We also had a good approximation for the variance of a random variable in the sample variance `\(s^2\)`, however, we will write it slightly different this time to make other models easy to understand. -- - Our estimator for the variance will be denoted as: `$$\hat{\sigma}^2_0 = \frac{1}{n} \sum_i \sum_j \left(y_{ij}-\hat{\mu}\right)^2$$` -- - If you look at your previous notes, you will see that we replaced the value of the mean `\(\bar{y}\)` with our **estimator** `\(\hat{\mu}\)` and that we changed from `\(s^2\)` to `\(\hat{\sigma}_0^2\)`. -- - This is because "0" indicates that this is an estimator for the variance of the **Null Model**. --- ## The Null Model - Together, we refer to our new estimators `\(\hat{\mu}\)` and `\(\hat{\sigma}_0^2\)` as: -- 1. Model Prediction: `\(\hat{\mu}\)` -- 1. Mean Squared Error: `\(\hat{\sigma}_0^2\)` -- - When using this type of statistical models (like we will most of this class) we will also be interested on the Sum of Squared Errors, which is denoted as: `$$SSE_0 = \sum_i \sum_j \left(y_{ij}-\hat{\mu}\right)^2$$` -- - Again we have added as "0" to make it clear that this is the Sum of Squared Errors associated to the **Null Model**. --- - Form teams of 3 and calculate the model prediction, the sum of squared errors and the mean squared error of the **Null Model** for the smokers data: <div id="htmlwidget-cf67810fac9b111a3846" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-cf67810fac9b111a3846">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["1","2","3","4","5","6","7","8"],[1,2,3,4,5,6,7,8],[78.3,80,73.6,77.4,68,64.4,70.8,65.5],["non_smoker","non_smoker","non_smoker","non_smoker","smoker","smoker","smoker","smoker"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>id<\/th>\n <th>lung_capacity<\/th>\n <th>smoke_status<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":8,"columnDefs":[{"className":"dt-right","targets":[1,2]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ## Smokers data: Null Model .can-edit.key-likes[ - Prediction: ] .can-edit.key-likes[ - SSE: ] .can-edit.key-likes[ - mean Squared Error: ]